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Trazado el gráfico de la función/ sin(2*x+1)^(8)/(x)

Gráfico de la función y = sin(2*x+1)^(8)/(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          8         
       sin (2*x + 1)
f(x) = -------------
             x
f(x)=sin⁡8(2x+1)xf{\left(x \right)} = \frac{\sin^{8}{\left(2 x + 1 \right)}}{x} 
f = sin(2*x + 1)^8/x
Gráfico de la función
02468-8-6-4-2-1010-1010
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
sin⁡8(2x+1)x=0\frac{\sin^{8}{\left(2 x + 1 \right)}}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−12x_{1} = - \frac{1}{2}
x2=−12+π2x_{2} = - \frac{1}{2} + \frac{\pi}{2}
Solución numérica
x1=12.0615523593694x_{1} = 12.0615523593694
x2=−9.93025078054588x_{2} = -9.93025078054588
x3=−66.4484631710123x_{3} = -66.4484631710123
x4=82.7588551365849x_{4} = 82.7588551365849
x5=−99.4778752930652x_{5} = -99.4778752930652
x6=84.3181874491064x_{6} = 84.3181874491064
x7=48.1983402251652x_{7} = 48.1983402251652
x8=−58.6193757176184x_{8} = -58.6193757176184
x9=−39.7685900729079x_{9} = -39.7685900729079
x10=−74.3065043690709x_{10} = -74.3065043690709
x11=70.1900718388325x_{11} = 70.1900718388325
x12=43.5070776543419x_{12} = 43.5070776543419
x13=−2.0781390617709x_{13} = -2.0781390617709
x14=−60.176912149219x_{14} = -60.176912149219
x15=−16.1934721944657x_{15} = -16.1934721944657
x16=23.0773619948886x_{16} = 23.0773619948886
x17=26.2066085561393x_{17} = 26.2066085561393
x18=98.4467846625904x_{18} = 98.4467846625904
x19=40.334736116208x_{19} = 40.334736116208
x20=35.6509384373175x_{20} = 35.6509384373175
x21=38.7452943448145x_{21} = 38.7452943448145
x22=−75.907719978107x_{22} = -75.907719978107
x23=−27.2273416939306x_{23} = -27.2273416939306
x24=34.0533485067427x_{24} = 34.0533485067427
x25=18.3430115752606x_{25} = 18.3430115752606
x26=76.4753979800327x_{26} = 76.4753979800327
x27=57.6424025358945x_{27} = 57.6424025358945
x28=−68.0410327593829x_{28} = -68.0410327593829
x29=−63.354806286356x_{29} = -63.354806286356
x30=−24.0574572376745x_{30} = -24.0574572376745
x31=4.21487400936572x_{31} = 4.21487400936572
x32=−80.6020417338525x_{32} = -80.6020417338525
x33=−83.752059271432x_{33} = -83.752059271432
x34=−97.9035822422468x_{34} = -97.9035822422468
x35=−25.6404138014377x_{35} = -25.6404138014377
x36=−41.363371831441x_{36} = -41.363371831441
x37=−71.2085828331969x_{37} = -71.2085828331969
x38=−31.9221425644025x_{38} = -31.9221425644025
x39=−61.7603245597903x_{39} = -61.7603245597903
x40=85.9014385526118x_{40} = 85.9014385526118
x41=−46.0492445613531x_{41} = -46.0492445613531
x42=−47.6321282879908x_{42} = -47.6321282879908
x43=−93.1987569558512x_{43} = -93.1987569558512
x44=100.028717010425x_{44} = 100.028717010425
x45=32.4718130944684x_{45} = 32.4718130944684
x46=65.4980873319397x_{46} = 65.4980873319397
x47=19.9261052500872x_{47} = 19.9261052500872
x48=−3.64867469102178x_{48} = -3.64867469102178
x49=24.610899760378x_{49} = 24.610899760378
x50=79.6338415702186x_{50} = 79.6338415702186
x51=62.3264609409385x_{51} = 62.3264609409385
x52=−38.1851938980327x_{52} = -38.1851938980327
x53=78.036927197878x_{53} = 78.036927197878
x54=−85.3462188211258x_{54} = -85.3462188211258
x55=−55.4946680332775x_{55} = -55.4946680332775
x56=92.1818072275963x_{56} = 92.1818072275963
x57=49.7815500246322x_{57} = 49.7815500246322
x58=68.5939098887096x_{58} = 68.5939098887096
x59=−36.6410507787002x_{59} = -36.6410507787002
x60=−11.5113930964754x_{60} = -11.5113930964754
x61=76.4551227467172x_{61} = 76.4551227467172
x62=54.46346633568x_{62} = 54.46346633568
x63=−52.314966256059x_{63} = -52.314966256059
x64=−17.7768558336054x_{64} = -17.7768558336054
x65=−82.1686361096965x_{65} = -82.1686361096965
x66=−44.4572427630439x_{66} = -44.4572427630439
x67=−33.5030469013085x_{67} = -33.5030469013085
x68=−49.218085772617x_{68} = -49.218085772617
x69=−19.3718969330877x_{69} = -19.3718969330877
x70=93.7648759018355x_{70} = 93.7648759018355
x71=−69.6238381109649x_{71} = -69.6238381109649
x72=60.736429575218x_{72} = 60.736429575218
x73=46.602406779266x_{73} = 46.602406779266
x74=10.4801380966139x_{74} = 10.4801380966139
x75=−2.06565042532466x_{75} = -2.06565042532466
x76=41.9178865758869x_{76} = 41.9178865758869
x77=13.6594136313178x_{77} = 13.6594136313178
x78=87.4890148256822x_{78} = 87.4890148256822
x79=63.9096642404996x_{79} = 63.9096642404996
x80=5.7981187773955x_{80} = 5.7981187773955
x81=−96.2980539037501x_{81} = -96.2980539037501
x82=−77.486276800206x_{82} = -77.486276800206
x83=82.727600721034x_{83} = 82.727600721034
x84=−6.80009000754028x_{84} = -6.80009000754028
x85=90.5854209493139x_{85} = 90.5854209493139
x86=−91.6155437918681x_{86} = -91.6155437918681
x87=56.0451380715854x_{87} = 56.0451380715854
x88=−30.3234324819537x_{88} = -30.3234324819537
x89=71.7732190788116x_{89} = 71.7732190788116
x90=−88.4397021726468x_{90} = -88.4397021726468
x91=−90.0328220283044x_{91} = -90.0328220283044
x92=27.789866101788x_{92} = 27.789866101788
x93=−53.9143152585877x_{93} = -53.9143152585877
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en sin(2*x + 1)^8/x.
sin⁡8(0⋅2+1)0\frac{\sin^{8}{\left(0 \cdot 2 + 1 \right)}}{0}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
16sin⁡7(2x+1)cos⁡(2x+1)x−sin⁡8(2x+1)x2=0\frac{16 \sin^{7}{\left(2 x + 1 \right)} \cos{\left(2 x + 1 \right)}}{x} - \frac{\sin^{8}{\left(2 x + 1 \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−39.7690415814106x_{1} = -39.7690415814106
x2=44.2669893704748x_{2} = 44.2669893704748
x3=−83.7521092873697x_{3} = -83.7521092873697
x4=92.180126112679x_{4} = 92.180126112679
x5=74.1124038660448x_{5} = 74.1124038660448
x6=−16.1985985143369x_{6} = -16.1985985143369
x7=49.7757257662834x_{7} = 49.7757257662834
x8=40.336793162601x_{8} = 40.336793162601
x9=76.4601213216483x_{9} = 76.4601213216483
x10=54.4686723668636x_{10} = 54.4686723668636
x11=−31.9197914147873x_{11} = -31.9197914147873
x12=27.7842773228085x_{12} = 27.7842773228085
x13=70.1886099012152x_{13} = 70.1886099012152
x14=88.249638354871x_{14} = 88.249638354871
x15=15.9914072668152x_{15} = 15.9914072668152
x16=−71.9707986653973x_{16} = -71.9707986653973
x17=−99.4711782805252x_{17} = -99.4711782805252
x18=−61.7605753228223x_{18} = -61.7605753228223
x19=−79.8248230209888x_{19} = -79.8248230209888
x20=−91.6107854724706x_{20} = -91.6107854724706
x21=−93.9620488632621x_{21} = -93.9620488632621
x22=−9.92833487169884x_{22} = -9.92833487169884
x23=59.9751375325493x_{23} = 59.9751375325493
x24=85.8988650413369x_{24} = 85.8988650413369
x25=−68.0421030359746x_{25} = -68.0421030359746
x26=−97.8925874929779x_{26} = -97.8925874929779
x27=78.0378906643915x_{27} = 78.0378906643915
x28=−38.1900904311503x_{28} = -38.1900904311503
x29=41.9157442133507x_{29} = 41.9157442133507
x30=4.21402137509175x_{30} = 4.21402137509175
x31=26.2055546761109x_{31} = 26.2055546761109
x32=−82.1730790172552x_{32} = -82.1730790172552
x33=56.0463169921838x_{33} = 56.0463169921838
x34=32.4772279170461x_{34} = 32.4772279170461
x35=−47.6292673192466x_{35} = -47.6292673192466
x36=30.1294911842192x_{36} = 30.1294911842192
x37=−60.1815810258627x_{37} = -60.1815810258627
x38=−2.06736574465965x_{38} = -2.06736574465965
x39=−17.777508108349x_{39} = -17.777508108349
x40=−24.0589560466668x_{40} = -24.0589560466668
x41=37.9836872856169x_{41} = 37.9836872856169
x42=66.2583722504124x_{42} = 66.2583722504124
x43=−13.8495123959454x_{43} = -13.8495123959454
x44=−5.99257254408229x_{44} = -5.99257254408229
x45=98.4515767826663x_{45} = 98.4515767826663
x46=100.029466114261x_{46} = 100.029466114261
x47=−63.3418850336411x_{47} = -63.3418850336411
x48=−53.9110872390394x_{48} = -53.9110872390394
x49=−75.9021061862936x_{49} = -75.9021061862936
x50=−25.6377544056686x_{50} = -25.6377544056686
x51=93.7585737823487x_{51} = 93.7585737823487
x52=−90.0336781637965x_{52} = -90.0336781637965
x53=8.13553870136171x_{53} = 8.13553870136171
x54=84.3198428328977x_{54} = 84.3198428328977
x55=−55.4884078858686x_{55} = -55.4884078858686
x56=12.0631874938417x_{56} = 12.0631874938417
x57=−96.3062821196124x_{57} = -96.3062821196124
x58=−41.3517449056536x_{58} = -41.3517449056536
x59=79.6227585650116x_{59} = 79.6227585650116
x60=−42.9209407951649x_{60} = -42.9209407951649
x61=−49.9794590374976x_{61} = -49.9794590374976
x62=−27.9878191636024x_{62} = -27.9878191636024
x63=52.121077382413x_{63} = 52.121077382413
x64=62.3283170984472x_{64} = 62.3283170984472
x65=−57.833525584199x_{65} = -57.833525584199
x66=81.9664259031444x_{66} = 81.9664259031444
x67=63.9073065448004x_{67} = 63.9073065448004
x68=22.2751438331967x_{68} = 22.2751438331967
x69=96.1036489281767x_{69} = 96.1036489281767
x70=90.5939688866853x_{70} = 90.5939688866853
x71=48.1970842850461x_{71} = 48.1970842850461
x72=−74.3150164941039x_{72} = -74.3150164941039
x73=−77.479799595138x_{73} = -77.479799595138
x74=−35.8420454727256x_{74} = -35.8420454727256
x75=−64.9083490646998x_{75} = -64.9083490646998
x76=−46.0505290330457x_{76} = -46.0505290330457
x77=−33.4970017468524x_{77} = -33.4970017468524
x78=−91.6122792204312x_{78} = -91.6122792204312
x79=−85.3311398318543x_{79} = -85.3311398318543
x80=−69.620775460956x_{80} = -69.620775460956
x81=−14.6285220385696x_{81} = -14.6285220385696
x82=34.0547473579879x_{82} = 34.0547473579879
x83=18.3452698401653x_{83} = 18.3452698401653
x84=19.9241779101437x_{84} = 19.9241779101437
x85=−3.64621911498531x_{85} = -3.64621911498531
x86=71.7671571612764x_{86} = 71.7671571612764
x87=68.6027670409628x_{87} = 68.6027670409628
Signos de extremos en los puntos:
(-39.76904158141065, -2.04736992557217e-24)

(44.26698937047478, 0.0225900166367254)

(-83.7521092873697, -2.21105461291369e-32)

(92.180126112679, 2.6433147377044e-20)

(74.11240386604479, 0.0134929796245852)

(-16.19859851433693, -9.34394416832983e-16)

(49.77572576628337, 6.23019252809181e-16)

(40.33679316260098, 3.47618151980688e-19)

(76.46012132164826, 1.31592365898015e-16)

(54.4686723668636, 2.40906721454624e-16)

(-31.919791414787305, -3.99238519887195e-19)

(27.784277322808464, 8.80045812691833e-16)

(70.18860990121516, 1.2832287376428e-20)

(88.24963835487101, 0.0113314684609724)

(15.991407266815209, 0.0625297626958592)

(-71.97079866539733, -0.0138944822347557)

(-99.47117828052522, -5.55217872132451e-16)

(-61.76057532282229, -1.19601560764386e-26)

(-79.82482302098884, -0.0125274007511097)

(-91.6107854724706, -5.58711720043359e-19)

(-93.9620488632621, -0.0106425758307417)

(-9.928334871698837, -6.6056896155952e-19)

(59.975137532549304, 0.0166735033428155)

(85.89886504133695, 1.29493111972596e-18)

(-68.04210303597458, -1.64878580212547e-21)

(-97.89258749297791, -2.98651298299217e-20)

(78.03789066439153, 6.2028378655372e-22)

(-38.1900904311503, -2.93965846685813e-16)

(41.915744213350685, 6.41984546558787e-19)

(4.2140213750917495, 3.06293995046942e-21)

(26.20555467611087, 2.67726954519918e-21)

(-82.17307901725518, -7.2193769344955e-17)

(56.046316992183776, 4.25956058279443e-21)

(32.477227917046115, 5.20465630337809e-16)

(-47.629267319246594, -3.75796201611612e-18)

(30.129491184219223, 0.033189501338378)

(-60.1815810258627, -1.36935625387709e-16)

(-2.0673657446596483, -2.37540662491414e-18)

(-17.777508108348954, -8.66511864804943e-23)

(-24.058956046666815, -6.77612742585817e-20)

(37.983687285616874, 0.0263268061002243)

(66.25837225041244, 0.0150923786226206)

(-13.84951239594544, -0.0721988265227307)

(-5.992572544082289, -0.166800653073985)

(98.45157678266634, 7.71830195807645e-17)

(100.02946611426103, 6.51714983801152e-23)

(-63.3418850336411, -4.14383873405849e-16)

(-53.91108723903943, -3.18803324057877e-19)

(-75.90210618629362, -1.74117756545954e-19)

(-25.637754405668616, -3.98296091022217e-18)

(93.75857378234872, 5.02891877571803e-16)

(-90.03367816379654, -2.10280399824959e-22)

(8.135538701361709, 0.122888477080844)

(84.31984283289775, 3.00939730588116e-20)

(-55.48840788586863, -7.00381066535669e-16)

(12.063187493841742, 2.23655238148007e-19)

(-96.30628211961239, -1.38588955793292e-15)

(-41.35174490565356, -1.36566785866734e-15)

(79.62275856501158, 1.52159198850622e-15)

(-42.920940795164874, -3.75949803953792e-16)

(-49.979459037497605, -0.0200080946081427)

(-27.98781916360241, -0.0357291165562022)

(52.121077382412984, 0.0191859857581803)

(62.328317098447194, 1.00368507968916e-19)

(-57.83352558419898, -0.0172909280277426)

(81.96642590314441, 0.0122000887963)

(63.90730654480042, 8.8633560505496e-19)

(22.275143833196726, 0.0448916746688043)

(96.10364892817674, 0.010405414564261)

(90.59396888668533, 1.40221639031e-15)

(48.19708428504606, 5.81062717119916e-21)

(-74.3150164941039, -1.9374169490801e-15)

(-77.47979959513805, -6.02001747531436e-16)

(-35.84204547272561, -0.02789985404233)

(-64.9083490646998, -4.39197113354359e-18)

(-46.0505290330457, -1.03976663154432e-20)

(-33.49700174685237, -9.51824931894044e-16)

(-91.61227922043123, -5.30189780391124e-18)

(-85.3311398318543, -5.77031401800756e-17)

(-69.62077546095601, -4.31472695975375e-18)

(-14.628522038569551, -5.45691565261464e-16)

(34.054747357987914, 2.61926303534964e-20)

(18.345269840165344, 1.58911968440939e-18)

(19.92417791014365, 5.89492466247743e-19)

(-3.6462191149853123, -1.47346276579505e-17)

(71.76715716127636, 5.37287813086988e-16)

(68.6027670409628, 1.91727709501533e-15)


Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
x1=−71.9707986653973x_{1} = -71.9707986653973
x2=−79.8248230209888x_{2} = -79.8248230209888
x3=−93.9620488632621x_{3} = -93.9620488632621
x4=−13.8495123959454x_{4} = -13.8495123959454
x5=−5.99257254408229x_{5} = -5.99257254408229
x6=−49.9794590374976x_{6} = -49.9794590374976
x7=−27.9878191636024x_{7} = -27.9878191636024
x8=−57.833525584199x_{8} = -57.833525584199
x9=−35.8420454727256x_{9} = -35.8420454727256
Puntos máximos de la función:
x9=44.2669893704748x_{9} = 44.2669893704748
x9=74.1124038660448x_{9} = 74.1124038660448
x9=88.249638354871x_{9} = 88.249638354871
x9=15.9914072668152x_{9} = 15.9914072668152
x9=59.9751375325493x_{9} = 59.9751375325493
x9=30.1294911842192x_{9} = 30.1294911842192
x9=37.9836872856169x_{9} = 37.9836872856169
x9=66.2583722504124x_{9} = 66.2583722504124
x9=8.13553870136171x_{9} = 8.13553870136171
x9=52.121077382413x_{9} = 52.121077382413
x9=81.9664259031444x_{9} = 81.9664259031444
x9=22.2751438331967x_{9} = 22.2751438331967
x9=96.1036489281767x_{9} = 96.1036489281767
Decrece en los intervalos
[−5.99257254408229,8.13553870136171]\left[-5.99257254408229, 8.13553870136171\right]
Crece en los intervalos
(−∞,−93.9620488632621]\left(-\infty, -93.9620488632621\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
2(−16sin⁡2(2x+1)+112cos⁡2(2x+1)−16sin⁡(2x+1)cos⁡(2x+1)x+sin⁡2(2x+1)x2)sin⁡6(2x+1)x=0\frac{2 \left(- 16 \sin^{2}{\left(2 x + 1 \right)} + 112 \cos^{2}{\left(2 x + 1 \right)} - \frac{16 \sin{\left(2 x + 1 \right)} \cos{\left(2 x + 1 \right)}}{x} + \frac{\sin^{2}{\left(2 x + 1 \right)}}{x^{2}}\right) \sin^{6}{\left(2 x + 1 \right)}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=31.8810251466497x_{1} = 31.8810251466497
x2=−5.81181361942402x_{2} = -5.81181361942402
x3=−65.8682560111894x_{3} = -65.8682560111894
x4=12.0645003620035x_{4} = 12.0645003620035
x5=88.0689544469536x_{5} = 88.0689544469536
x6=−47.6268758159073x_{6} = -47.6268758159073
x7=−17.77805413387x_{7} = -17.77805413387
x8=−24.0602316441611x_{8} = -24.0602316441611
x9=66.0776880746672x_{9} = 66.0776880746672
x10=70.1874060399061x_{10} = 70.1874060399061
x11=−53.9103372583214x_{11} = -53.9103372583214
x12=−46.0516221114031x_{12} = -46.0516221114031
x13=85.8966719980706x_{13} = 85.8966719980706
x14=−39.7694195916574x_{14} = -39.7694195916574
x15=38.1643727131207x_{15} = 38.1643727131207
x16=−90.034406763783x_{16} = -90.034406763783
x17=7.35679011035367x_{17} = 7.35679011035367
x18=−83.7521511651907x_{18} = -83.7521511651907
x19=26.2046741475496x_{19} = 26.2046741475496
x20=19.9225374775483x_{20} = 19.9225374775483
x21=92.1787805664853x_{21} = 92.1787805664853
x22=84.3212276595072x_{22} = 84.3212276595072
x23=48.1960389413519x_{23} = 48.1960389413519
x24=90.0011247701253x_{24} = 90.0011247701253
x25=22.0944548359114x_{25} = 22.0944548359114
x26=18.3471585730361x_{26} = 18.3471585730361
x27=−91.6095502183051x_{27} = -91.6095502183051
x28=36.2321698721516x_{28} = 36.2321698721516
x29=−14.0302099726304x_{29} = -14.0302099726304
x30=−27.8071321596035x_{30} = -27.8071321596035
x31=−43.8768687936668x_{31} = -43.8768687936668
x32=−2.0688274402156x_{32} = -2.0688274402156
x33=−87.8595236350033x_{33} = -87.8595236350033
x34=60.1558218429737x_{34} = 60.1558218429737
x35=−25.6355314216225x_{35} = -25.6355314216225
x36=−99.4643157968925x_{36} = -99.4643157968925
x37=−7.74518286427396x_{37} = -7.74518286427396
x38=53.8725757459175x_{38} = 53.8725757459175
x39=−26.5976438596002x_{39} = -26.5976438596002
x40=−61.7607852791307x_{40} = -61.7607852791307
x41=14.2397015088219x_{41} = 14.2397015088219
x42=−80.0055070055054x_{42} = -80.0055070055054
x43=56.0473142096494x_{43} = 56.0473142096494
x44=137.729270218678x_{44} = 137.729270218678
x45=−29.7393614699179x_{45} = -29.7393614699179
x46=40.338513614353x_{46} = 40.338513614353
x47=62.3298696878929x_{47} = 62.3298696878929
x48=−3.64416818202166x_{48} = -3.64416818202166
x49=−93.7813649961376x_{49} = -93.7813649961376
x50=65.477259439363x_{50} = 65.477259439363
x51=100.030103421995x_{51} = 100.030103421995
x52=44.0863044331053x_{52} = 44.0863044331053
x53=−58.0142099510687x_{53} = -58.0142099510687
x54=58.2236428399471x_{54} = 58.2236428399471
x55=34.0559148343214x_{55} = 34.0559148343214
x56=−90.036381879809x_{56} = -90.036381879809
x57=−9.92693618279064x_{57} = -9.92693618279064
x58=−49.7987743966485x_{58} = -49.7987743966485
x59=63.9052983467241x_{59} = 63.9052983467241
x60=−21.8849998819882x_{60} = -21.8849998819882
x61=67.047860023559x_{61} = 67.047860023559
x62=82.1471098658718x_{62} = 82.1471098658718
x63=−96.3176846288348x_{63} = -96.3176846288348
x64=80.2149381484301x_{64} = 80.2149381484301
x65=−36.0227311297403x_{65} = -36.0227311297403
x66=−68.0430138335567x_{66} = -68.0430138335567
x67=4.21330824333771x_{67} = 4.21330824333771
x68=−31.918486398718x_{68} = -31.918486398718
x69=−69.6182153620487x_{69} = -69.6182153620487
x70=−64.8987061318351x_{70} = -64.8987061318351
x71=−71.7901145832379x_{71} = -71.7901145832379
x72=96.2843327816804x_{72} = 96.2843327816804
x73=78.0387092477765x_{73} = 78.0387092477765
x74=16.1721013434242x_{74} = 16.1721013434242
x75=41.9139201353585x_{75} = 41.9139201353585
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−(2(−16sin⁡2(2x+1)+112cos⁡2(2x+1)−16sin⁡(2x+1)cos⁡(2x+1)x+sin⁡2(2x+1)x2)sin⁡6(2x+1)x)=−∞\lim_{x \to 0^-}\left(\frac{2 \left(- 16 \sin^{2}{\left(2 x + 1 \right)} + 112 \cos^{2}{\left(2 x + 1 \right)} - \frac{16 \sin{\left(2 x + 1 \right)} \cos{\left(2 x + 1 \right)}}{x} + \frac{\sin^{2}{\left(2 x + 1 \right)}}{x^{2}}\right) \sin^{6}{\left(2 x + 1 \right)}}{x}\right) = -\infty
lim⁡x→0+(2(−16sin⁡2(2x+1)+112cos⁡2(2x+1)−16sin⁡(2x+1)cos⁡(2x+1)x+sin⁡2(2x+1)x2)sin⁡6(2x+1)x)=∞\lim_{x \to 0^+}\left(\frac{2 \left(- 16 \sin^{2}{\left(2 x + 1 \right)} + 112 \cos^{2}{\left(2 x + 1 \right)} - \frac{16 \sin{\left(2 x + 1 \right)} \cos{\left(2 x + 1 \right)}}{x} + \frac{\sin^{2}{\left(2 x + 1 \right)}}{x^{2}}\right) \sin^{6}{\left(2 x + 1 \right)}}{x}\right) = \infty
- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
[96.2843327816804,∞)\left[96.2843327816804, \infty\right)
Convexa en los intervalos
(−∞,−87.8595236350033]\left(-\infty, -87.8595236350033\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(sin⁡8(2x+1)x)=0\lim_{x \to -\infty}\left(\frac{\sin^{8}{\left(2 x + 1 \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡8(2x+1)x)=0\lim_{x \to \infty}\left(\frac{\sin^{8}{\left(2 x + 1 \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(2*x + 1)^8/x, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡8(2x+1)x2)=0\lim_{x \to -\infty}\left(\frac{\sin^{8}{\left(2 x + 1 \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡8(2x+1)x2)=0\lim_{x \to \infty}\left(\frac{\sin^{8}{\left(2 x + 1 \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
sin⁡8(2x+1)x=−sin⁡8(2x−1)x\frac{\sin^{8}{\left(2 x + 1 \right)}}{x} = - \frac{\sin^{8}{\left(2 x - 1 \right)}}{x}
- No
sin⁡8(2x+1)x=sin⁡8(2x−1)x\frac{\sin^{8}{\left(2 x + 1 \right)}}{x} = \frac{\sin^{8}{\left(2 x - 1 \right)}}{x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор